Index
Charts, bundle cocycles, and metrics additions
New knowls added while annotating a conversation about charts, principal-bundle cocycles, gauge transformations, and Riemannian metrics.
Core idea
This index records the concepts added while knowlifying a conversation that began with coordinate charts in the category of smooth manifolds and developed the relation between smooth atlases, principal-bundle cocycles, gauge transformations, frame bundles, and Riemannian metrics.
The full uncommitted batch affects 33 knowl files relative to develop: 20 additions, 12 modifications, and one deletion. The additions comprise 17 mathematical concept knowls, this index, and two development-only documents. The production-content comparison excludes those two testing documents and therefore contains 31 knowl changes.
New definitions and bridge results
- Smooth compatibility of charts and atlases
- Maximal smooth atlas
- Smooth structure
- Coordinate transition map
- Coordinate chart as an isomorphism of smooth manifolds
- Open submanifold
- G-torsor
- Fiber coordinates
- Smooth G-valued Čech 1-cocycle
- Nonabelian Čech H1 and principal bundles
- Čech cocycle groupoid
- Pointed set
- Tangent-bundle cocycle from coordinate changes
- Coordinate transformation law for a Riemannian metric
- Locally finite family
- Support of a function
- Symmetric positive-definite matrix
Duplicate consolidation
The former fiber-bundles/fiber-of-a-map-preimage-fiber entry duplicated the existing Fiber of a map knowl. All content in this batch now uses the canonical entry.
Modified existing knowls
Development-only documents
The development build also adds the knowlified transcript posts/charts-bundle-cocycles-and-metrics and the temporary audit index posts/rewrite-candidates-index. They remain under testing/ and are not part of production content.
Existing backbone
The annotated conversation also uses these pre-existing knowls:
- Category of smooth manifolds
- Smooth coordinate chart
- Smooth atlas
- Fiber of a map
- Partition of unity subordinate to an open cover
- Principal G-bundle
- Bundle atlas
- Local trivialization
- Equivalence of cocycles
- Gauge transformation
- Frame bundle of a manifold
- Reduction of structure group
- Riemannian manifold
- Principal connection
- Curvature 2-form of a principal connection